## Quotient Rule (In what order do you choose which to differentiate)

1. The problem statement, all variables and given/known data

The Derivative of x-2/(x^2-1)

Can you show me how to get the derivative using the quotient rule and tell me if the order matters? I always thought that the order never mattered until I got a wrong graph because it seems I got the wrong derivative.

2. Relevant equations

3. The attempt at a solution

So which one is correct?
If I plug in 0.1 on each of these one gives me a + answer and the other -

I got -x^2+4x-1/(x^2-1)^2
and
x^2-4x+1/(x^2-1)^2

At first I thought these where the same but then as I was plugging in random point to see if it increased or decreased at the critical points I did not get the same results. It seems that ive used the quotient rule correctly but the answer changes depending on which you choose first to differentiate.
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 Recognitions: Gold Member Science Advisor Staff Emeritus You can evaluate a compound expression in any order you want, as long as you do it correctly. What do you mean by "order matters"? I have a suspicion you are comparing two different formulas, rather than comparing two different ways to evaluate a single formula.
 Well you know how we have (f/g) what is (f/g)' ? is it f'g-g'f/g^2 or g'f-f'g/g^2 OR does it even matter? I thought it didn't matter until I came to this problem

Recognitions:
Homework Help

## Quotient Rule (In what order do you choose which to differentiate)

 Quote by Jani08 Well you know how we have (f/g) what is (f/g)' ? is it f'g-g'f/g^2 or g'f-f'g/g^2 OR does it even matter? I thought it didn't matter until I came to this problem
(f'g-g'f)/g^2

Recognitions:
Gold Member
Staff Emeritus
 Quote by Jani08 Well you know how we have (f/g) what is (f/g)' ? is it f'g-g'f/g^2 or g'f-f'g/g^2 OR does it even matter? I thought it didn't matter until I came to this problem
Well, the two formulas
$$\frac{f'g - g'f}{g^2}$$
and
$$\frac{g'f - f'g}{g^2}$$
are formally different. Of course, that doesn't mean they are actually unequal -- have you thought about trying to prove/disprove their equality? It would be one of those "prove or disprove the following identity" kind of problems. I guess you have already done the "disproof", since you found a counterexample. Do you remember why you thought it didn't matter?

Incidentally, what you wrote is technically wrong: the meaning of
$$f'g - g'f/g^2$$
is
$$f'g - \frac{g'f}{g^2}$$
and not
$$\frac{f'g - g'f}{g^2}$$
Even if you know what you mean, you really should add parentheses when talking to other people:
$$(f'g - g'f)/g^2$$
because they might not realize that you are abusing notation. Actually, I would advise using parentheses even when you're writing just for yourself -- I've seen many people make mistakes because they got confused on how things were grouped.
 Mentor Note the parentheses that gb7nash added. When you write x-2/(x^2-1), people would interpret this as x - (2/(x^2 - 1)), which is probably not what you intended. Also, (f'g-g'f)/g^2 and (g'f - f'g)/g^2 have opposite signs.
 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus I suspect you were confused because the order doesn't matter with the product rule. That's because in the product rule, you're adding the terms, and addition is commutative. If you differentiate f first, you get f'g+g'f, and if you differentiate g first, you get g'f+f'g. Since the order doesn't matter for addition, those two expressions are equal. With the quotient rule, however, there's a subtraction rather than an addition. As you know, the order does matter for subtraction: a-b isn't the same as b-a. So for the quotient rule, you need to remember that you differentiate f first and g second. If you can't remember that, just write f/g as f(g-1) and use the product and chain rules to rederive the quotient rule.
 Hurkyl thanks for tips, I guess I should type it the same way I type it in my graphing calculator. Vela you got it spot on, now that I look at it the way of properties it does make sense. Hah I finally got it...after a whole year doing this thing..I was always just "hoping" I had the right order never really tackled it. Thanks everyone!